Which Equation Describes How The Parent Function, $y=x^3$, Is Vertically Stretched By A Factor Of 4?A. $y=x^3+4$ B. $y=(x+4)^3$ C. $y=4x^3$ D. $y=x^4$

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When dealing with parent functions, understanding how they can be transformed is crucial in mathematics. One common transformation is the vertical stretch, which involves multiplying the parent function by a factor. In this article, we will explore how the parent function y=x3y=x^3 is vertically stretched by a factor of 4.

What is a Vertical Stretch?

A vertical stretch is a transformation that involves multiplying the parent function by a factor. This means that the function is stretched vertically, making it taller or shorter. The factor used for the vertical stretch determines the extent of the transformation.

Parent Function: y=x3y=x^3

The parent function y=x3y=x^3 is a cubic function that represents a curve. This function has a single turning point, known as the vertex, and is symmetric about the origin.

Vertical Stretch by a Factor of 4

To vertically stretch the parent function y=x3y=x^3 by a factor of 4, we need to multiply the function by 4. This means that the new function will have the same shape as the parent function, but it will be 4 times taller.

Equation for the Vertically Stretched Function

The equation for the vertically stretched function can be written as:

y=4x3y=4x^3

This equation represents the parent function y=x3y=x^3 stretched vertically by a factor of 4.

Comparing the Options

Let's compare the given options with the equation we derived:

  • A. y=x3+4y=x^3+4 - This equation adds 4 to the parent function, which is not a vertical stretch.
  • B. y=(x+4)3y=(x+4)^3 - This equation shifts the parent function horizontally by 4 units, which is not a vertical stretch.
  • C. y=4x3y=4x^3 - This equation is the same as the one we derived, which represents a vertical stretch by a factor of 4.
  • D. y=x4y=x^4 - This equation changes the exponent of the parent function, which is not a vertical stretch.

Conclusion

In conclusion, the equation that describes how the parent function y=x3y=x^3 is vertically stretched by a factor of 4 is:

y=4x3y=4x^3

This equation represents the parent function stretched vertically by a factor of 4, making it 4 times taller.

Key Takeaways

  • A vertical stretch involves multiplying the parent function by a factor.
  • The factor used for the vertical stretch determines the extent of the transformation.
  • The equation for the vertically stretched function can be written as y=4x3y=4x^3.

Practice Problems

  1. What is the equation for the vertically stretched function y=x2y=x^2 by a factor of 3?
  2. What is the equation for the vertically stretched function y=x4y=x^4 by a factor of 2?
  3. What is the equation for the vertically stretched function y=x3y=x^3 by a factor of 5?

Answer Key

  1. y=3x2y=3x^2
  2. y=2x4y=2x^4
  3. y=5x3y=5x^3

References

In the previous article, we explored how the parent function y=x3y=x^3 is vertically stretched by a factor of 4. In this article, we will answer some frequently asked questions (FAQs) about vertical stretches.

Q: What is a vertical stretch?

A: A vertical stretch is a transformation that involves multiplying the parent function by a factor. This means that the function is stretched vertically, making it taller or shorter.

Q: How do I determine the equation for a vertically stretched function?

A: To determine the equation for a vertically stretched function, you need to multiply the parent function by the factor used for the vertical stretch. For example, if the parent function is y=x3y=x^3 and the vertical stretch factor is 4, the equation for the vertically stretched function is y=4x3y=4x^3.

Q: What is the difference between a vertical stretch and a horizontal shift?

A: A vertical stretch involves multiplying the parent function by a factor, while a horizontal shift involves adding or subtracting a value from the parent function. For example, if the parent function is y=x3y=x^3 and the horizontal shift is 4, the equation for the shifted function is y=(x−4)3y=(x-4)^3.

Q: Can I apply multiple transformations to a function?

A: Yes, you can apply multiple transformations to a function. For example, if you want to apply a vertical stretch by a factor of 4 and a horizontal shift by 2 to the parent function y=x3y=x^3, the equation for the transformed function is y=4(x−2)3y=4(x-2)^3.

Q: How do I determine the factor used for a vertical stretch?

A: The factor used for a vertical stretch is usually given in the problem. If it's not given, you can use the information provided to determine the factor. For example, if the problem states that the function is stretched by a factor of 2, the factor used for the vertical stretch is 2.

Q: Can I apply a vertical stretch to any function?

A: Yes, you can apply a vertical stretch to any function. However, the factor used for the vertical stretch must be a positive number. If the factor is negative, it will result in a reflection of the function across the x-axis.

Q: How do I graph a vertically stretched function?

A: To graph a vertically stretched function, you can use the following steps:

  1. Graph the parent function.
  2. Identify the factor used for the vertical stretch.
  3. Multiply the parent function by the factor to get the equation for the vertically stretched function.
  4. Graph the vertically stretched function.

Q: What are some common mistakes to avoid when applying vertical stretches?

A: Some common mistakes to avoid when applying vertical stretches include:

  • Using a negative factor for the vertical stretch, which will result in a reflection of the function across the x-axis.
  • Forgetting to multiply the parent function by the factor used for the vertical stretch.
  • Not identifying the factor used for the vertical stretch.

Conclusion

In conclusion, vertical stretches are an important concept in mathematics that can help you understand how functions can be transformed. By following the steps outlined in this article, you can apply vertical stretches to functions and determine the equation for the transformed function.

Key Takeaways

  • A vertical stretch involves multiplying the parent function by a factor.
  • The factor used for the vertical stretch determines the extent of the transformation.
  • You can apply multiple transformations to a function.
  • The factor used for a vertical stretch must be a positive number.

Practice Problems

  1. What is the equation for the vertically stretched function y=x2y=x^2 by a factor of 3?
  2. What is the equation for the vertically stretched function y=x4y=x^4 by a factor of 2?
  3. What is the equation for the vertically stretched function y=x3y=x^3 by a factor of 5?

Answer Key

  1. y=3x2y=3x^2
  2. y=2x4y=2x^4
  3. y=5x3y=5x^3

References